Irreversible growth model for virus capsid assembly

Hexagonal Lattices of HIV Capsid Proteins Explored by Simulations Based on a Thermodynamically Consistent Model

HIV capsid proteins (CAs) may self-assemble into a variety of shapes under in vivo and in vitro conditions. Here, we employed simulations based on a residue-level coarse-grained (CG) model with full conformational flexibility to investigate hexagonal lattices, which are the underlying structural pattern for CA aggregations. Facilitated by enhanced sampling simulations to rigorously calculate CA dimerization and polymerization affinities, we calibrated our model to reproduce the experimentally measured affinities. Using the calibrated model, we performed unbiased simulations on several large systems consisting of 1512 CA subunits, allowing reversible binding and unbinding of the CAs in a thermodynamically consistent manner. In one simulation, a preassembled hexagonal CA sheet developed spontaneous curvatures reminiscent of those observed in experiments, and the edges of the sheet exhibited local curvatures larger than those of the interior. In other simulations starting with randomly distributed CAs at different concentrations, existing CA assemblies grew by binding free capsomeres to the edges and by merging with other assemblies. At high CA concentrations, rapid establishment of predominant aggregates was followed by much slower adjustments toward more regular hexagonal lattices, with increasing numbers of intact CA hexamers and pentamers being formed. Our approach of adapting a general CG model to specific systems by using experimental binding data represents a practical and effective strategy for simulating and elucidating intricate protein aggregations.

Cooperative Self-Assembly Process Involving Giant Toroidal Polyoxometalate as a Membrane Building Block in Nanoscale Vesicles

The self-assembly of organic amphiphilic species into various aggregates such as spherical or elongated micelles and cylinders up to the formation of lyotropic hexagonal or lamellar phases results from cooperative processes orchestrated by the hydrophobic effect, while those involving ionic inorganic polynuclear entities and nonionic organic components are still intriguing. Herein, we report on the supramolecular behavior of giant toroidal molybdenum blue-type polyoxometalate, namely, the {Mo154} species in the presence of n-octyl-β-glucoside (C8G1), widely used as a surfactant in biochemistry. Structural investigations were carried out using a set of complementary multiscale methods including single-crystal X-ray diffraction analysis supported by molecular modeling, small-angle X-ray scattering and cryo-TEM observations. In addition, liquid NMR, viscosimetry, surface tension measurement, and isothermal titration calorimetry provided further information to decipher the complex aggregation pathway. Elucidation of the assembly process reveals a rich scenario where the presence of the large {Mo154} anion disrupts the self-assembly of the C8G1, well-known to produce micelles, and induces striking successive phase transitions from fluid-to-gel and from gel-to-fluid. Herein, intimate organic-inorganic primary interactions arising from the superchaotropic nature of the {Mo154} lead to versatile nanoscopic hybrid C8G1-{Mo154} aggregates including crystalline discrete assemblies, smectic lamellar liquid crystals, and large uni- or multilamellar vesicles where the large torus {Mo154} acts a trans-membrane component.

Virus Assembly Pathways Inside a Host Cell

Simple RNA viruses self-assemble spontaneously and encapsulate their genome into a shell called the capsid. This process is mainly driven by the attractive electrostatics interaction between the positive charges on capsid proteins and the negative charges on the genome. Despite its importance and many decades of intense research, how the virus selects and packages its native RNA inside the crowded environment of a host cell cytoplasm in the presence of an abundance of nonviral RNA and other anionic polymers has remained a mystery. In this paper, we perform a series of simulations to monitor the growth of viral shells and find the mechanism by which cargo-coat protein interactions can impact the structure and stability of the viral shells. We show that coat protein subunits can assemble around a globular nucleic acid core by forming nonicosahedral cages, which have been recently observed in assembly experiments involving small pieces of RNA. We find that the resulting cages are strained and can easily be split into fragments along stress lines. This suggests that such metastable nonicosahedral intermediates could be easily reassembled into the stable native icosahedral shells if the larger wild-type genome becomes available, despite the presence of a myriad of nonviral RNAs.

Phase transitions in virology

Viruses have established relationships with almost every other living organism on Earth and at all levels of biological organization: from other viruses up to entire ecosystems. In most cases, they peacefully coexist with their hosts, but in most relevant cases, they parasitize them and induce diseases and pandemics, such as the AIDS and the most recent avian influenza and COVID-19 pandemic events, causing a huge impact on health, society, and economy. Viruses play an essential role in shaping the eco-evolutionary dynamics of their hosts, and have been also involved in some of the major evolutionary innovations either by working as vectors of genetic information or by being themselves coopted by the host into their genomes. Viruses can be studied at different levels of biological organization, from the molecular mechanisms of genome replication, gene expression and encapsidation, to global pandemics. All these levels are different and yet connected through the presence of threshold conditions allowing for the formation of a capsid, the loss of genetic information or epidemic spreading. These thresholds, as occurs with temperature separating phases in a liquid, define sharp qualitative types of behaviour. Thesephase transitionsare very well known in physics. They have been studied by means of simple, but powerful models able to capture their essential properties, allowing us to better understand them. Can the physics of phase transitions be an inspiration for our understanding of viral dynamics at different scales? Here we review well-known mathematical models of transition phenomena in virology. We suggest that the advantages of abstract, simplified pictures used in physics are also the key to properly understanding the origins and evolution of complexity in viruses. By means of several examples, we explore this multilevel landscape and how minimal models provide deep insights into a diverse array of problems. The relevance of these transitions in connecting dynamical patterns across scales and their evolutionary and clinical implications are outlined.

Role of combined cell membrane and wall mechanical properties regulated by polarity signals in cell budding

Budding yeast, Saccharomyces cerevisiae, serves as a prime biological model to study mechanisms underlying asymmetric growth. Previous studies have shown that prior to bud emergence, polarization of a conserved small GTPase Cdc42 must be established on the cell membrane of a budding yeast. Additionally, such polarization contributes to the delivery of cell wall remodeling enzymes and hydrolase from cytosol through the membrane, to change the mechanical properties of the cell wall. This leads to the hypothesis that Cdc42 and its associated proteins at least indirectly regulate cell surface mechanical properties. However, how the surface mechanical properties in the emerging bud are changed and whether such change is important are not well understood. To test several hypothesised mechanisms, a novel three-dimensional coarse-grained particle-based model has been developed which describes inhomogeneous mechanical properties of the cell surface. Model simulations predict alternation of the levels of stretching and bending stiffness of the cell surface in the bud region by the polarized Cdc42 signals is essential for initiating bud formation. Model simulations also suggest that bud shape depends strongly on the distribution of the polarized signaling molecules while the neck width of the emerging bud is strongly impacted by the mechanical properties of the chitin and septin rings. Moreover, the temporal change of the bud mechanical properties is shown to affect the symmetry of the bud shape. The 3D model of asymmetric cell growth can also be used for studying viral budding and other vegetative reproduction processes performed via budding, as well as detailed studies of cell growth.

How a Virus Circumvents Energy Barriers to Form Symmetric Shells

Previous self-assembly experiments on a model icosahedral plant virus have shown that, under physiological conditions, capsid proteins initially bind to the genome through an en masse mechanism and form nucleoprotein complexes in a disordered state, which raises the question as to how virions are assembled into a highly ordered structure in the host cell. Using small-angle X-ray scattering, we find out that a disorder-order transition occurs under physiological conditions upon an increase in capsid protein concentrations. Our cryo-transmission electron microscopy reveals closed spherical shells containing in vitro transcribed viral RNA even at pH 7.5, in marked contrast with the previous observations. We use Monte Carlo simulations to explain this disorder-order transition and find that, as the shell grows, the structures of disordered intermediates in which the distribution of pentamers does not belong to the icosahedral subgroups become energetically so unfavorable that the caps can easily dissociate and reassemble, overcoming the energy barriers for the formation of perfect icosahedral shells. In addition, we monitor the growth of capsids under the condition that the nucleation and growth is the dominant pathway and show that the key for the disorder-order transition in both en masse and nucleation and growth pathways lies in the strength of elastic energy compared to the other forces in the system including protein-protein interactions and the chemical potential of free subunits. Our findings explain, at least in part, why perfect virions with icosahedral order form under different conditions including physiological ones.

Measurements of the self-assembly kinetics of individual viral capsids around their RNA genome

Self-assembly is a process in which functional nanoscale structures build themselves, driven by Brownian motion and interactions between components. The term was originally coined to describe the formation of a viral capsid, the protein shell that protects the genome of a virus. Despite decades of study, how capsids self-assemble has remained a mystery, because there were no methods to measure the assembly kinetics of individual capsids. We surmount this obstacle using a sensitive microscopy technique based on laser interferometry. The measurements show that a small nucleus of proteins must form on the viral RNA before the capsid assembles. These results might help researchers design strategies to stop the assembly of pathogenic viruses or to build synthetic nanostructures.

Self-assembly is widely used by biological systems to build functional nanostructures, such as the protein capsids of RNA viruses. But because assembly is a collective phenomenon involving many weakly interacting subunits and a broad range of timescales, measurements of the assembly pathways have been elusive. We use interferometric scattering microscopy to measure the assembly kinetics of individual MS2 bacteriophage capsids around MS2 RNA. By recording how many coat proteins bind to each of many individual RNA strands, we find that assembly proceeds by nucleation followed by monotonic growth. Our measurements reveal the assembly pathways in quantitative detail and also show their failure modes. We use these results to critically examine models of the assembly process.

A Multiscale Model for the Self-Assembly of Coat Proteins in Bacteriophage MS2

The self-assembly of viral capsids is an essential step to the formation of infectious viruses. Elucidating the kinetic mechanisms of how a capsid or virus-like particle assembles could advance our knowledge about the viral lifecycle, as well as the general principles in self-assembly of biomaterials. However, current understanding of capsid assembly remains incomplete for many viruses due to the fact that the transient intermediates along the assembling pathways are experimentally difficult to be detected. In this paper, we constructed a new multiscale computational framework to simulate the self-assembly of virus-like particles. We applied our method to the coat proteins of bacteriophage MS2 as a specific model system. This virus-like particle of bacteriophage MS2 has a unique feature that its 90 sequence-identical dimers can be classified into two structurally various groups: one is the symmetric CC dimer, and the other is the asymmetric AB dimer. The homotypic interactions between AB dimers result in a 5-fold symmetric contact, while the heterotypic interactions between AB and CC dimers result in 6-fold symmetric contact. We found that the assembly can be described as a physical process of phase transition that is regulated by various factors such as concentration and specific stoichiometry between AB and CC dimers. Our simulations also demonstrate that heterotypic and homotypic interfaces play distinctive roles in modulating the assembling kinetics. The interaction between AB and CC dimers is much more dynamic than that between two AB dimers. We therefore suggest that the alternate growth of viral capsid through the heterotypic dimer interactions dominates the assembling pathways. This is, to the best of our knowledge, the first multiscale model to simulate the assembling process of coat proteins in bacteriophage MS2. The generality of this approach opens the door to its further applications in assembly of other viral capsids, virus-like particles, and novel drug delivery systems.

Mechanical stress relaxation in molecular self-assembly

Molecular self-assembly on a curved substrate leads to the spontaneous inclusion of topological defects in the growing bidimensional crystal, unlike assembly on a flat substrate. We propose in this work a quantitative mechanism for this phenomenon by using standard thin shell elasticity. The Gaussian curvature of the substrate induces large in-plane compressive stress as the surface grows, in particular at the rim of the assembly, and the addition of a single defect relaxes this mechanical stress. We found out that the value of azimuthal stress at the rim of the assembly determines the preferred directions for defect nucleation. These results are also discussed as a function of different defect combinations, like dislocations and grain boundaries or scars. In particular, the elastic model permits us to compare quantitatively the ability of various defects to relax mechanical stress. Moreover, these findings allow us to understand the progressive building-up of the typical disclination and grain boundary pattern observed for ground states of large 2D spherical crystals.

The equilibrium structure of self-assembled protein nano-cages

Understanding how highly symmetric, robust, monodisperse protein nano-cages self-assemble can have major applications in various areas of bio-nanotechnology, such as drug delivery, biomedical imaging and gene therapy. We develop a model to investigate the assembly of protein subunits into the structures with different sizes and symmetries. Using Monte Carlo simulation, we obtain global minimum energy structures. Our results suggest that the physical properties including the spontaneous curvature, flexibility and bending rigidity of coat proteins are sufficient to predict the size, symmetry and shape selectivity of the assembly products. Further, on a thermodynamic basis, we discuss the polymorphism of nano-cages observed in assembly experiments.

Molecular jenga: the percolation phase transition (collapse) in virus capsids

Virus capsids are polymeric protein shells that protect the viral cargo. About half of known virus families have icosahedral capsids that self-assemble from tens to thousands of subunits. Capsid disassembly is critical to the lifecycles of many viruses yet is poorly understood. Here, we apply a graph and percolation theory to examine the effect of removing capsid subunits on capsid stability and fragmentation. Based on the structure of the icosahedral capsid of hepatitis B virus (HBV), we constructed a graph of rhombic subunits arranged with icosahedral symmetry. Though our approach neglects dependence on energetics, time, and molecular detail, it quantitatively predicts a percolation phase transition consistent with recent in vitro studies of HBV capsid dissociation. While the stability of the capsid graph followed a gradual quadratic decay, the rhombic tiling abruptly fragmented when we removed more than 25% of the 120 subunits, near the percolation threshold observed experimentally. This threshold may also affect results of capsid assembly, which also experimentally produces a preponderance of 90 mer intermediates, as the intermediate steps in these reactions are reversible and can thus resemble dissociation. Application of percolation theory to understanding capsid association and dissociation may prove a general approach to relating virus biology to the underlying biophysics of the virus particle.

Robust nonequilibrium pathways to microcompartment assembly

Cyanobacteria sequester photosynthetic enzymes into microcompartments which facilitate the conversion of carbon dioxide into sugars. Geometric similarities between these structures and self-assembling viral capsids have inspired models that posit microcompartments as stable equilibrium arrangements of the constituent proteins. Here we describe a different mechanism for microcompartment assembly, one that is fundamentally nonequilibrium and yet highly reliable. This pathway is revealed by simulations of a molecular model resolving the size and shape of a cargo droplet and the extent and topography of an elastic shell. The resulting metastable microcompartment structures closely resemble those of carboxysomes, with a narrow size distribution and faceted shells. The essence of their assembly dynamics can be understood from a simpler mathematical model that combines elements of classical nucleation theory with continuum elasticity. These results highlight important control variables for achieving nanoscale encapsulation in general and for modulating the size and shape of carboxysomes in particular.

Molecular dynamics study of T = 3 capsid assembly

Molecular dynamics simulation is used to model the self-assembly of polyhedral shells containing 180 trapezoidal particles that correspond to the T = 3 virus capsid. Three kinds of particle, differing only slightly in shape, are used to account for the effect of quasi-equivalence. Bond formation between particles is reversible and an explicit atomistic solvent is included. Under suitable conditions the simulations are able to produce complete shells, with the majority of unused particles remaining as monomers, and practically no other clusters. There are also no incorrectly assembled clusters. The simulations reveal details of intermediate structures along the growth pathway, information that is relevant for interpreting experiment.

Kinetic constraints on self-assembly into closed supramolecular structures

Many biological and synthetic systems exploit self-assembly to generate highly intricate closed supramolecular architectures, ranging from self-assembling cages to viral capsids. The fundamental design principles that control the structural determinants of the resulting assemblies are increasingly well-understood, but much less is known about the kinetics of such assembly phenomena and it remains a key challenge to elucidate how these systems can be engineered to assemble in an efficient manner and avoid kinetic trapping. We show here that simple scaling laws emerge from a set of kinetic equations describing the self-assembly of identical building blocks into closed supramolecular structures and that this scaling behavior provides general rules that determine efficient assembly in these systems. Using this framework, we uncover the existence of a narrow range of parameter space that supports efficient self-assembly and reveal that nature capitalizes on this behavior to direct the reliable assembly of viral capsids on biologically relevant timescales.

Viral self-assembly pathway and mechanical stress relaxation

The final shape of a virus is dictated by the self-assembly pathway of its constituents. Using standard thin-shell elasticity, we highlight the prominent role of the viral shell's spontaneous curvature in determining the assembly pathway. In particular, we demonstrate that the mechanical stress inherent to the growth of a curved surface can be relaxed in two different ways in the early steps of assembly, depending on the value of the spontaneous curvature of the surface. This important result explains why most viral shells have either a compact shape with icosahedral symmetry or an elongated shape lacking this symmetry.

Reaction-diffusion basis of retroviral infectivity

Retrovirus particle (virion) infectivity requires diffusion and clustering of multiple transmembrane envelope proteins (Env₃) on the virion exterior, yet is triggered by protease-dependent degradation of a partially occluding, membrane-bound Gag polyprotein lattice on the virion interior. The physical mechanism underlying such coupling is unclear and only indirectly accessible via experiment. Modelling stands to provide insight but the required spatio-temporal range far exceeds current accessibility by all-atom or even coarse-grained molecular dynamics simulations. Nor do such approaches account for chemical reactions, while conversely, reaction kinetics approaches handle neither diffusion nor clustering. Here, a recently developed multiscale approach is considered that applies an ultra-coarse-graining scheme to treat entire proteins at near-single particle resolution, but which also couples chemical reactions with diffusion and interactions. A model is developed of Env₃ molecules embedded in a truncated Gag lattice composed of membrane-bound matrix proteins linked to capsid subunits, with freely diffusing protease molecules. Simulations suggest that in the presence of Gag but in the absence of lateral lattice-forming interactions, Env₃ diffuses comparably to Gag-absent Env₃ Initial immobility of Env₃ is conferred through lateral caging by matrix trimers vertically coupled to the underlying hexameric capsid layer. Gag cleavage by protease vertically decouples the matrix and capsid layers, induces both matrix and Env₃ diffusion, and permits Env₃ clustering. Spreading across the entire membrane surface reduces crowding, in turn, enhancing the effect and promoting infectivity.This article is part of the themed issue 'Multiscale modelling at the physics-chemistry-biology interface'.

Assembly of the most topologically regular two-dimensional micro and nanocrystals with spherical, conical, and tubular shapes

We consider how to control the extension of curvature-induced defects in the hexagonal order covering different curved surfaces. In these frames we propose a physical mechanism for improving structures of two-dimensional spherical colloidal crystals (SCCs). For any SCC comprising of about 300 or less particles the mechanism transforms all extended topological defects (ETDs) in the hexagonal order into the point disclinations. Perfecting the structure is carried out by successive cycles of the particle implantation and subsequent relaxation of the crystal. The mechanism is potentially suitable for obtaining colloidosomes with better selective permeability. Our approach enables modeling the most topologically regular tubular and conical two-dimensional nanocrystals including various possible polymorphic forms of the HIV viral capsid. Different HIV-like shells with an arbitrary number of structural units (SUs) and desired geometrical parameters are easily formed. Faceting of the obtained structures is performed by minimizing the suggested elastic energy.

Computational virology: From the inside out

Viruses typically pack their genetic material within a protein capsid. Enveloped viruses also have an outer membrane made up of a lipid bilayer and membrane-spanning glycoproteins. X-ray diffraction and cryoelectron microscopy provide high resolution static views of viral structure. Molecular dynamics (MD) simulations may be used to provide dynamic insights into the structures of viruses and their components. There have been a number of simulations of viral capsids and (in some cases) of the inner core of RNA or DNA packaged within them. These simulations have generally focussed on the structural integrity and stability of the capsid and/or on the influence of the nucleic acid core on capsid stability. More recently there have been a number of simulation studies of enveloped viruses, including HIV-1, influenza A, and dengue virus. These have addressed the dynamic behaviour of the capsid, the matrix, and/or of the outer envelope. Analysis of the dynamics of the lipid bilayer components of the envelopes of influenza A and of dengue virus reveals a degree of biophysical robustness, which may contribute to the stability of virus particles in different environments. Significant computational challenges need to be addressed to aid simulation of complex viruses and their membranes, including the need to integrate structural data from a range of sources to enable us to move towards simulations of intact virions. This article is part of a Special Issue entitled: Membrane Proteins edited by J.C. Gumbart and Sergei Noskov.

The Robust Assembly of Small Symmetric Nanoshells

Highly symmetric nanoshells are found in many biological systems, such as clathrin cages and viral shells. Many studies have shown that symmetric shells appear in nature as a result of the free-energy minimization of a generic interaction between their constituent subunits. We examine the physical basis for the formation of symmetric shells, and by using a minimal model, demonstrate that these structures can readily grow from the irreversible addition of identical subunits. Our model of nanoshell assembly shows that the spontaneous curvature regulates the size of the shell while the mechanical properties of the subunit determine the symmetry of the assembled structure. Understanding the minimum requirements for the formation of closed nanoshells is a necessary step toward engineering of nanocontainers, which will have far-reaching impact in both material science and medicine.

Allosteric Control of Icosahedral Capsid Assembly

During the life cycle of a virus, viral proteins and other components self-assemble to form an ordered protein shell called a capsid. This assembly process is subject to multiple competing constraints, including the need to form a thermostable shell while avoiding kinetic traps. It has been proposed that viral assembly satisfies these constraints through allosteric regulation, including the interconversion of capsid proteins among conformations with different propensities for assembly. In this article, we use computational and theoretical modeling to explore how such allostery affects the assembly of icosahedral shells. We simulate assembly under a wide range of protein concentrations, protein binding affinities, and two different mechanisms of allosteric control. We find that above a threshold strength of allosteric control, assembly becomes robust over a broad range of subunit binding affinities and concentrations, allowing the formation of highly thermostable capsids. Our results suggest that allostery can significantly shift the range of protein binding affinities that lead to successful assembly and thus should be taken into account in models that are used to estimate interaction parameters from experimental data.

HIV Capsid Assembly, Mechanism, and Structure

The HIV genome materials are encaged by a proteinaceous shell called the capsid, constructed from ∼1000-1500 copies of the capsid proteins. Because its stability and integrity are critical to the normal life cycle and infectivity of the virus, the HIV capsid is a promising antiviral drug target. In this paper, we review the studies shaping our understanding of the structure and dynamics of the capsid proteins and various forms of their assemblies, as well as the assembly mechanism.

Mechanism of polymorphism and curvature of HIV capsid assemblies probed by 3D simulations with a novel coarse grain model

BACKGROUND

During the maturation process, HIV capsid proteins self-assemble into polymorphic capsids. The strong polymorphism precludes high resolution structural characterization under in vivo conditions. In spite of the determination of structural models for various in vitro assemblies of HIV capsid proteins, the assembly mechanism is still not well-understood.

METHODS

We report 3D simulations of HIV capsid proteins by a novel coarse grain model that captures the backbone of the rigid segments in the protein accurately. The effects of protein dynamics on assembly are emulated by a static ensemble of subunits in conformations derived from molecular dynamics simulation.

RESULTS

We show that HIV capsid proteins robustly assemble into hexameric lattices in a range of conditions where trimers of dimeric subunits are the dominant oligomeric intermediates. Variations of hexameric lattice curvatures are observed in simulations with subunits of variable inter-domain orientations mimicking the conformation distribution in solution. Simulations with subunits based on pentameric structural models lead to assembly of sharp curved structures resembling the tips of authentic HIV capsids, along a distinct pathway populated by tetramers and pentamers with the characteristic quasi-equivalency of viral capsids.

CONCLUSIONS

Our results suggest that the polymorphism assembly is triggered by the inter-domain dynamics of HIV capsid proteins in solution. The assembly of highly curved structures arises from proteins in conformation with a highly specific inter-domain orientation.

SIGNIFICANCE

Our work proposes a mechanism of HIV capsid assembly based on available structural data, which can be readily verified. Our model can be applied to other large biomolecular assemblies.

The Role of Packaging Sites in Efficient and Specific Virus Assembly

During the life cycle of many single-stranded RNA viruses, including many human pathogens, a protein shell called the capsid spontaneously assembles around the viral genome. Understanding the mechanisms by which capsid proteins selectively assemble around the viral RNA amidst diverse host RNAs is a key question in virology. In one proposed mechanism, short sequences (packaging sites) within the genomic RNA promote rapid and efficient assembly through specific interactions with the capsid proteins. In this work, we develop a coarse-grained particle-based computational model for capsid proteins and RNA that represents protein-RNA interactions arising both from nonspecific electrostatics and from specific packaging site interactions. Using Brownian dynamics simulations, we explore how the efficiency and specificity of assembly depend on solution conditions (which control protein-protein and nonspecific protein-RNA interactions) and the strength and number of packaging sites. We identify distinct regions in parameter space in which packaging sites lead to highly specific assembly via different mechanisms and others in which packaging sites lead to kinetic traps. We relate these computational predictions to in vitro assays for specificity in which cognate viral RNAs compete against non-cognate RNAs for assembly by capsid proteins.

Mechanisms of virus assembly

Viruses are nanoscale entities containing a nucleic acid genome encased in a protein shell called a capsid and in some cases are surrounded by a lipid bilayer membrane. This review summarizes the physics that govern the processes by which capsids assemble within their host cells and in vitro. We describe the thermodynamics and kinetics for the assembly of protein subunits into icosahedral capsid shells and how these are modified in cases in which the capsid assembles around a nucleic acid or on a lipid bilayer. We present experimental and theoretical techniques used to characterize capsid assembly, and we highlight aspects of virus assembly that are likely to receive significant attention in the near future.

Modeling Viral Capsid Assembly

I present a review of the theoretical and computational methodologies that have been used to model the assembly of viral capsids. I discuss the capabilities and limitations of approaches ranging from equilibrium continuum theories to molecular dynamics simulations, and I give an overview of some of the important conclusions about virus assembly that have resulted from these modeling efforts. Topics include the assembly of empty viral shells, assembly around single-stranded nucleic acids to form viral particles, and assembly around synthetic polymers or charged nanoparticles for nanotechnology or biomedical applications. I present some examples in which modeling efforts have promoted experimental breakthroughs, as well as directions in which the connection between modeling and experiment can be strengthened.

Simulations show that virus assembly and budding are facilitated by membrane microdomains

For many viruses, assembly and budding occur simultaneously during virion formation. Understanding the mechanisms underlying this process could promote biomedical efforts to block viral propagation and enable use of capsids in nanomaterials applications. To this end, we have performed molecular dynamics simulations on a coarse-grained model that describes virus assembly on a fluctuating lipid membrane. Our simulations show that the membrane can promote association of adsorbed subunits through dimensional reduction, but it also introduces thermodynamic and kinetic effects that can inhibit complete assembly. We find several mechanisms by which membrane microdomains, such as lipid rafts, reduce these effects, and thus, enhance assembly. We show how these predicted mechanisms can be experimentally tested. Furthermore, the simulations demonstrate that assembly and budding depend crucially on the system dynamics via multiple timescales related to membrane deformation, protein diffusion, association, and adsorption onto the membrane.

Molecular dynamics simulation: a tool for exploration and discovery using simple models

Emergent phenomena share the fascinating property of not being obvious consequences of the design of the system in which they appear. This characteristic is no less relevant when attempting to simulate such phenomena, given that the outcome is not always a foregone conclusion. The present survey focuses on several simple model systems that exhibit surprisingly rich emergent behavior, all studied by molecular dynamics (MD) simulation.The examples are taken from the disparate fields of fluid dynamics, granular matter and supramolecular self-assembly. In studies of fluids modeled at the detailed microscopic level using discrete particles, the simulations demonstrate that complex hydrodynamic phenomena in rotating and convecting fluids—the Taylor–Couette and Rayleigh–Bénard instabilities—cannot only be observed within the limited length and time scales accessible to MD, but even allow quantitative agreement to be achieved. Simulation of highly counter-intuitive segregation phenomena in granular mixtures, again using MD methods, but now augmented by forces producing damping and friction, leads to results that resemble experimentally observed axial and radial segregation in the case of a rotating cylinder and to a novel form of horizontal segregation in a vertically vibrated layer. Finally, when modeling self-assembly processes analogous to the formation of the polyhedral shells that package spherical viruses, simulation of suitably shaped particles reveals the ability to produce complete, error-free assembly and leads to the important general observation that reversible growth steps contribute to the high yield. While there are limitations to the MD approach, both computational and conceptual, the results offer a tantalizing hint of the kinds of phenomena that can be explored and what might be discovered when sufficient resources are brought to bear on a problem.

Pathways for virus assembly around nucleic acids

Understanding the pathways by which viral capsid proteins assemble around their genomes could identify key intermediates as potential drug targets. In this work, we use computer simulations to characterize assembly over a wide range of capsid protein-protein interaction strengths and solution ionic strengths. We find that assembly pathways can be categorized into two classes, in which intermediates are either predominantly ordered or disordered. Our results suggest that estimating the protein-protein and the protein-genome binding affinities may be sufficient to predict which pathway occurs. Furthermore, the calculated phase diagrams suggest that knowledge of the dominant assembly pathway and its relationship to control parameters could identify optimal strategies to thwart or redirect assembly to block infection. Finally, analysis of simulation trajectories suggests that the two classes of assembly pathways can be distinguished in single-molecule fluorescence correlation spectroscopy or bulk time-resolved small-angle X-ray scattering experiments.

Using Markov state models to study self-assembly

Markov state models (MSMs) have been demonstrated to be a powerful method for computationally studying intramolecular processes such as protein folding and macromolecular conformational changes. In this article, we present a new approach to construct MSMs that is applicable to modeling a broad class of multi-molecular assembly reactions. Distinct structures formed during assembly are distinguished by their undirected graphs, which are defined by strong subunit interactions. Spatial inhomogeneities of free subunits are accounted for using a recently developed Gaussian-based signature. Simplifications to this state identification are also investigated. The feasibility of this approach is demonstrated on two different coarse-grained models for virus self-assembly. We find good agreement between the dynamics predicted by the MSMs and long, unbiased simulations, and that the MSMs can reduce overall simulation time by orders of magnitude.

Viral genome structures are optimal for capsid assembly

Understanding how virus capsids assemble around their nucleic acid (NA) genomes could promote efforts to block viral propagation or to reengineer capsids for gene therapy applications. We develop a coarse-grained model of capsid proteins and NAs with which we investigate assembly dynamics and thermodynamics. In contrast to recent theoretical models, we find that capsids spontaneously ‘overcharge’; that is, the negative charge of the NA exceeds the positive charge on capsid. When applied to specific viruses, the optimal NA lengths closely correspond to the natural genome lengths. Calculations based on linear polyelectrolytes rather than base-paired NAs underpredict the optimal length, demonstrating the importance of NA structure to capsid assembly. These results suggest that electrostatics, excluded volume, and NA tertiary structure are sufficient to predict assembly thermodynamics and that the ability of viruses to selectively encapsidate their genomic NAs can be explained, at least in part, on a thermodynamic basis.

DOI:http://dx.doi.org/10.7554/eLife.00632.001

Viruses are infectious agents made up of proteins and a genome made of DNA or RNA. Upon infecting a host cell, viruses hijack the cell’s gene expression machinery and force it to produce copies of the viral genome and proteins, which then assemble into new viruses that can eventually infect other host cells. Because assembly is an essential step in the viral life cycle, understanding how this process occurs could significantly advance the fight against viral diseases.

In many viral families, a protein shell called a capsid forms around the viral genome during the assembly process. However, capsids can also assemble around nucleic acids in solution, indicating that a host cell is not required for their formation. Since capsid proteins are positively charged, and nucleic acids are negatively charged, electrostatic interactions between the two are thought to have an important role in capsid assembly. However, it is unclear how structural features of the viral genome affect assembly, and why the negative charge on viral genomes is actually far greater than the positive charge on capsids. These questions are difficult to address experimentally because most of the intermediates that form during virus assembly are too short-lived to be imaged.

Here, Perlmutter et al. have used state of the art computational methods and advances in graphical processing units (GPUs) to produce the most realistic model of capsid assembly to date. They showed that the stability of the complex formed between the nucleic acid and the capsid depends on the length of the viral genome. Yield was highest for genomes within a certain range of lengths, and capsids that assembled around longer or shorter genomes tended to be malformed.

Perlmutter et al. also explored how structural features of the virus—including base-pairing between viral nucleic acids, and the size and charge of the capsid—determine the optimal length of the viral genome. When they included structural data from real viruses in their simulations and predicted the optimal lengths for the viral genome, the results were very similar to those seen in existing viruses. This indicates that the structure of the viral genome has been optimized to promote packaging into capsids. Understanding this relationship between structure and packaging will make it easier to develop antiviral agents that thwart or misdirect virus assembly, and could aid the redesign of viruses for use in gene therapy and drug delivery.

DOI:http://dx.doi.org/10.7554/eLife.00632.002

Impact of the topology of viral RNAs on their encapsulation by virus coat proteins

Single-stranded RNAs of simple viruses seem to be topologically more compact than other types of single-stranded RNA. It has been suggested that this has an evolutionary purpose: more compact structures are more easily encapsulated in the limited space that the cavity of the virus capsid offers. We employ a simple Flory theory to calculate the optimal amount of polymers confined in a viral shell. We find that the free energy gain or more specifically the efficiency of RNA encapsidation increases substantially with topological compactness. We also find that the optimal length of RNA encapsidated in a capsid increases with the degree of branching of the genome even though this effect is very weak. Further, we show that if the structure of the branching of the polymer is allowed to anneal, the optimal loading increases substantially.

Theoretical studies on assembly, physical stability and dynamics of viruses

All matter has to obey the general laws of physics and living matter is not an exception. Viruses have not only learnt how to cope with them, but have managed to use them for their own survival. In this chapter we will review some of the exciting physics behind viruses and discuss simple physical models that can shed some light on different aspects of the viral life cycle and viral properties. In particular, we will focus on how the structure and shape of the capsid, its assembly and stability, and the entry and exit of viral particles and their genomes can be understood using fundamental physics theories.

Molecular dynamics simulation of reversibly self-assembling shells in solution using trapezoidal particles

The self-assembly of polyhedral shells, each constructed from 60 trapezoidal particles, is simulated using molecular dynamics. The spatial organization of the component particles in this shell is similar to the capsomer proteins forming the capsid of a T=1 virus. Growth occurs in the presence of an atomistic solvent and, under suitable conditions, achieves a high yield of complete shells. The simulations provide details of the structure and lifetime of the particle clusters that appear as intermediate states along the growth pathway, and the nature of the transitions between them. In certain respects the growth of size-60 shells from trapezoidal particles resembles the growth of icosahedral shells from triangular particles studied previously, with reversible bonding playing a major role in avoiding incorrect assembly, although the details differ due to particle shape and bond organization. The strong preference for maximal bonding exhibited by the triangular particle clusters is also apparent for trapezoidal particles, but this is now confined to early growth and is less pronounced as shells approach completion along a variety of pathways.

Unclosed HIV-1 capsids suggest a curled sheet model of assembly

The RNA genome of retroviruses is encased within a protein capsid. To gather insight into the assembly and function of this capsid, we used electron cryotomography to image human immunodeficiency virus (HIV) and equine infectious anemia virus (EIAV) particles. While the majority of viral cores appeared closed, a variety of unclosed structures including rolled sheets, extra flaps, and cores with holes in the tip were also seen. Simulations of nonequilibrium growth of elastic sheets recapitulated each of these aberrations and further predicted the occasional presence of seams, for which tentative evidence was also found within the cryotomograms. To test the integrity of viral capsids in vivo, we observed that ~25% of cytoplasmic HIV complexes captured by TRIM5α had holes large enough to allow internal green fluorescent protein (GFP) molecules to escape. Together, these findings suggest that HIV assembly at least sometimes involves the union in space of two edges of a curling sheet and results in a substantial number of unclosed forms.

Interaction of virions with membrane glycolipids

Cellular membranes contain various lipids including glycolipids (GLs). The hydrophilic head groups of GLs extend from the membrane into the aqueous environment outside the cell where they act as recognition sites for specific interactions. The first steps of interaction of virions with cells often include contacts with GLs. To clarify the details of such contacts, we have used the total internal reflection fluorescence microscopy to explore the interaction of individual unlabelled virus-like particles (or, more specifically, norovirus protein capsids), which are firmly bound to a lipid bilayer, and fluorescent vesicles containing glycosphingolipids (these lipids form a subclass of GLs). The corresponding binding kinetics were earlier found to be kinetically limited, while the detachment kinetics were logarithmic over a wide range of time. Here, the detachment rate is observed to dramatically decrease with increasing concentration of glycosphingolipids from 1% to 8%. This effect has been analytically explained by using a generic model describing the statistics of bonds in the contact area between a virion and a lipid membrane. Among other factors, the model takes the formation of GL domains into account. Our analysis indicates that in the system under consideration, such domains, if present, have a characteristic size smaller than the contact area between the vesicle and the virus-like particle.

Mechanisms of kinetic trapping in self-assembly and phase transformation

In self-assembly processes, kinetic trapping effects often hinder the formation of thermodynamically stable ordered states. In a model of viral capsid assembly and in the phase transformation of a lattice gas, we show how simulations in a self-assembling steady state can be used to identify two distinct mechanisms of kinetic trapping. We argue that one of these mechanisms can be adequately captured by kinetic rate equations, while the other involves a breakdown of theories that rely on cluster size as a reaction coordinate. We discuss how these observations might be useful in designing and optimising self-assembly reactions.

Encapsulation of a polymer by an icosahedral virus

The coat proteins of many viruses spontaneously form icosahedral capsids around nucleic acids or other polymers. Elucidating the role of the packaged polymer in capsid formation could promote biomedical efforts to block viral replication and enable use of capsids in nanomaterials applications. To this end, we perform Brownian dynamics on a coarse-grained model that describes the dynamics of icosahedral capsid assembly around a flexible polymer. We identify several mechanisms by which the polymer plays an active role in its encapsulation, including cooperative polymer-protein motions. These mechanisms are related to experimentally controllable parameters such as polymer length, protein concentration and solution conditions. Furthermore, the simulations demonstrate that assembly mechanisms are correlated with encapsulation efficiency, and we present a phase diagram that predicts assembly outcomes as a function of experimental parameters. We anticipate that our simulation results will provide a framework for designing in vitro assembly experiments on single-stranded RNA virus capsids.

Mechanisms of capsid assembly around a polymer

Capsids of many viruses assemble around nucleic acids or other polymers. Understanding how the properties of the packaged polymer affect the assembly process could promote biomedical efforts to prevent viral assembly or nanomaterials applications that exploit assembly. To this end, we simulate on a lattice the dynamical assembly of closed, hollow shells composed of several hundred to 1000 subunits, around a flexible polymer. We find that assembly is most efficient at an optimum polymer length that scales with the surface area of the capsid; polymers that are significantly longer than optimal often lead to partial-capsids with unpackaged polymer "tails" or a competition between multiple partial-capsids attached to a single polymer. These predictions can be tested with bulk experiments in which capsid proteins assemble around homopolymeric RNA or synthetic polyelectrolytes. We also find that the polymer can increase the net rate of subunit accretion to a growing capsid both by stabilizing the addition of new subunits and by enhancing the incoming flux of subunits; the effects of these processes may be distinguishable with experiments that monitor the assembly of individual capsids.

Understanding the concentration dependence of viral capsid assembly kinetics--the origin of the lag time and identifying the critical nucleus size

The kinetics for the assembly of viral proteins into a population of capsids can be measured in vitro with size exclusion chromatography or dynamic light scattering, but extracting mechanistic information from these studies is challenging. For example, it is not straightforward to determine the critical nucleus size or the elongation time (the time required for a nucleus to grow to completion). In this work, we study theoretical and computational models for capsid assembly to show that the critical nucleus size can be determined from the concentration dependence of the assembly half-life and that the elongation time is revealed by the length of the lag phase. Furthermore, we find that the system becomes kinetically trapped when nucleation becomes fast compared to elongation. Implications of this constraint for determining elongation mechanisms from experimental assembly data are discussed.

Studies of reversible capsid shell growth

Results from molecular dynamics simulations of simple, structured particles capable of self-assembling into polyhedral shells are described. The analysis focuses on the growth histories of individual shells in the presence of an explicit solvent and the nature of the events along their growth pathways; the results provide further evidence of the importance of reversibility in the assembly process. The underlying goal of this approach is the modeling of virus capsid growth, a phenomenon at the submicroscopic scale that, despite its importance, is little understood.

Role of protein interactions in defining HIV-1 viral capsid shape and stability: a coarse-grained analysis

Coarse-grained models of the HIV-1 CA dimer are constructed based on all-atom molecular dynamics simulations. Coarse-grained representations of the capsid shell, which is composed of approximately 1500 copies of CA proteins, are constructed and their stability is examined. A key interaction between carboxyl and hexameric amino terminal domains is shown to generate the curvature of the capsid shell. It is demonstrated that variation of the strength of this interaction for different subunits in the lattice can cause formation of asymmetric, conical-shaped closed capsid shells, and it is proposed that variations, in the structure of the additional carboxyl-amino terminal binding interface during self-assembly, are important aspects of capsid cone formation. These results are in agreement with recent structural studies of the capsid hexamer subunit, which suggest that variability in the binding interface is a cause of the differences in subunit environments that exist in a conical structure.

Coarse-grained protein-protein stiffnesses and dynamics from all-atom simulations

Large protein assemblies, such as virus capsids, may be coarse-grained as a set of rigid units linked by generalized (rotational and stretching) harmonic springs. We present an ab initio method to obtain the elastic parameters and overdamped dynamics for these springs from all-atom molecular-dynamics simulations of one pair of units at a time. The computed relaxation times of this pair give a consistency check for the simulation, and we can also find the corrective force needed to null systematic drifts. As a first application we predict the stiffness of an HIV capsid layer and the relaxation time for its breathing mode.

Nonequilibirum assembly, retroviruses, and conical structures

We study the spontaneous assembly of viral shells composed of several identical subunits under nonequilibrium conditions. We find that within the basic continuum elasticity framework, the nonequilibrium assembly process is able to predict the formation of structures pertinent to retroviruses. Our minimal model of assembly yields a unified one-dimensional phase diagram in which the appearance of spherical, irregular, conical and cylindrical structures of retroviruses is seen to be governed by the spontaneous curvature of protein subunits.

Mechanisms of size control and polymorphism in viral capsid assembly

We simulate the assembly dynamics of icosahedral capsids from subunits that interconvert between different conformations (or quasi-equivalent states). The simulations identify mechanisms by which subunits form empty capsids with only one morphology but adaptively assemble into different icosahedral morphologies around nanoparticle cargoes with varying sizes, as seen in recent experiments with brome mosaic virus (BMV) capsid proteins. Adaptive cargo encapsidation requires moderate cargo-subunit interaction strengths; stronger interactions frustrate assembly by stabilizing intermediates with incommensurate curvature. We compare simulation results to experiments with cowpea chlorotic mottle virus empty capsids and BMV capsids assembled on functionalized nanoparticles and suggest new cargo encapsidation experiments. Finally, we find that both empty and templated capsids maintain the precise spatial ordering of subunit conformations seen in the crystal structure even if interactions that preserve this arrangement are favored by as little as the thermal energy, consistent with experimental observations that different subunit conformations are highly similar.

Model of human immunodeficiency virus budding and self-assembly: role of the cell membrane

Budding from the plasma membrane of the host cell is an indispensable step in the life cycle of the human immunodeficiency virus (HIV), which belongs to a large family of enveloped RNA viruses, retroviruses. Unlike regular enveloped viruses, retrovirus budding happens concurrently with the self-assembly of the main retrovirus protein subunits (called Gag protein after the name of the genetic material that codes for this protein: Group-specific AntiGen) into spherical virus capsids on the cell membrane. Led by this unique budding and assembly mechanism, we study the free energy profile of retrovirus budding, taking into account the Gag-Gag attraction energy and the membrane elastic energy. We find that if the Gag-Gag attraction is strong, budding always proceeds to completion. During early stage of budding, the zenith angle of partial budded capsids, alpha , increases with time as alpha proportional t1/2. However, if the Gag-Gag attraction is weak, a metastable state of partial budding appears. The zenith angle of these partially spherical capsids is given by alpha0 approximately (tau2/kappasigma)1/4 in a linear approximation, where kappa and sigma are the bending modulus and the surface tension of the membrane, and tau is a line tension of the capsid proportional to the strength of Gag-Gag attraction. Numerically, we find alpha0<0.3pi without any approximations. Using experimental parameters, we show that HIV budding and assembly always proceed to completion in normal biological conditions. On the other hand, by changing Gag-Gag interaction strength or membrane rigidity, it is relatively easy to tune it back and forth between complete budding and partial budding. Our model agrees reasonably well with experiments observing partial budding of retroviruses including HIV.

Role of reversibility in viral capsid growth: a paradigm for self-assembly

Self-assembly at submicroscopic scales is an important but little understood phenomenon. A prominent example is virus capsid growth, whose underlying behavior can be modeled using simple particles that assemble into polyhedral shells. Molecular dynamics simulation of shell formation in the presence of an atomistic solvent provides new insight into the self-assembly mechanism, notably that growth proceeds via a cascade of strongly reversible steps and, despite the large variety of possible intermediates, only a small fraction of highly bonded forms appear on the pathway.